The frequency of change in the selective pressures is one of the main factors driving evolution. It is generally accepted that constant environments select specialist organisms whereas changing environments favour generalists. The particular outcome achieved in either case also depends on the relative strength of the selective pressures and on the fitness costs of mutations across environments. RNA viruses are characterized by their high genetic diversity, which provides fast adaptation to environmental changes and helps them evade most antiviral treatments. Therefore, the study of the adaptive possibilities of RNA viruses is highly relevant for both basic and applied research. In this study we have evolved an RNA virus, the bacteriophage Qβ, under three different temperatures that either were kept constant or alternated periodically. The populations obtained were analyzed at the phenotypic and the genotypic level to characterize the evolutionary process followed by the virus in each case and the amount of convergent genetic changes attained. Finally, we also investigated the influence of the pre-existent genetic diversity on adaptation to high temperature. The main conclusions that arise from our results are: i) under periodically changing temperature conditions, evolution of bacteriophage Qβ is driven by the most stringent selective pressure, ii) there is a high degree of evolutionary convergence between replicated populations and also among populations evolved at different temperatures, iii) there are mutations specific of a particular condition, and iv) adaptation to high temperatures in populations differing in their pre-existent genetic diversity takes place through the selection of a common set of mutations.